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Conditional finite entropy balance with explicit boundary flux

Physical scope

Supplied C1 entropy current on a finite spacelike-cap slab and adopted local nonnegative production; slice-entropy change plus net outward side flux equals integrated production; zero net side flux is sufficient for the endpoint entropy inequality.

Statement

Use the smooth time-oriented Lorentzian geometry, positively oriented chart and product slab $K = [x_{-}^{0}, x_{+}^{0}] \times Ω$ with open spatial rectangle $Ω$ and compact closure $\overline{K}$ of Finite spacelike slab and normal flux charge, with nonempty rectangular factors, spacelike slices, future unit normal $n$ , induced volume $d V_{h}$ , and a $C^{1}$ current defined on a neighborhood of $\overline{K}$ . Let $S^{a}$ be the entropy current selected by Local entropy-current admissibility in a separately defined model and put $Π : = \nabla_{a} S^{a} \geq 0$ on $K$ . Define the proper slice entropy

$Q_{S} (x^{0}) : = \int_{Ω} [- g (S, n)] d V_{h}$

and let $Φ_{S} (B)$ be the net outward flux through the union $B$ of the six spatial side faces, with the sign convention of Finite spacelike slab and normal flux charge. Conditional on the exact current statement and hypotheses of the draft mathematical theorem Complete finite-slab production and outward-flux balance,

$Q_{S} (x_{+}^{0}) - Q_{S} (x_{-}^{0}) + Φ_{S} (B) = \int_{K} Π {v o l}_{g} \geq 0.$

Every term has units $J K^{- 1}$ in the length-coordinate convention of Thermodynamics-GR fields, currents, reference normalization and units. If the net outward side flux is zero, then the endpoint charges satisfy $Q_{S} (x_{+}^{0}) \geq Q_{S} (x_{-}^{0})$ . This is a deduction conditional on both the adopted local inequality and the actual draft supplier interface; it is not an observation, a proof approval of that mathematical supplier, or a generalized second law.

Facts & Assumptions

Given: The Lorentzian chart slab, spacelike slices, future normal, compact closure and $C^{1}$ current specified in Finite spacelike slab and normal flux charge; an entropy current $S$ satisfying the adopted local condition; and the SI length-coordinate convention $x^{0} = c t_{p h y s}$ .

[F1]

Thermodynamics-GR fields, currents, reference normalization and units: $S^{a}$ has components $J K^{- 1} m^{- 3}$ , the physical entropy flux is $c S$ , and $\nabla_{a} S^{a}$ has units $J K^{- 1} m^{- 4}$ .

[A1]

Local entropy-current admissibility in a separately defined model adopts $Π = \nabla_{a} S^{a} \geq 0$ pointwise for the separately specified smooth model. This is a physical premise, not a geometric theorem or a measured result.

[L1]

Finite spacelike slab and normal flux charge is the current draft definition: $K$ is a compactly bounded coordinate product slab with spacelike caps, $S$ is $C^{1}$ near its closure, $Q_{S}$ is the future-normal/proper-volume charge, and $Φ_{S} (B)$ is the signed net outward coordinate-face flux.

[L2]

Complete finite-slab production and outward-flux balance is the current draft mathematical theorem. Its exact current statement is $Q_{J} (x_{+}^{0}) - Q_{J} (x_{-}^{0}) + Φ_{J} (B) = \int_{K} (\nabla_{a} J^{a}) {v o l}_{g}$ for the slab and $C^{1}$ current of [L1]. Its proof and manifest snapshot remain unresolved upstream; this use is explicitly conditional and does not treat it as proved.

Proof

technique · direct conditional application of the finite-slab balance, with positivity supplied by the adopted physical inequality

1.1givenA1L1algebra

By [A1], $Π \geq 0$ throughout $K$ . The oriented Lorentzian volume density is positive in the supplied chart and $K$ is a bounded Jordan slab, so its Riemann integral is defined and $\int_{K} Π {v o l}_{g} \geq 0$ . This is conditional on the smooth current and slab hypotheses in [L1].

2.1givenF1L1L2step 1.1

Apply [L2] to the current $J = S$ . By [L1], the resulting charge is exactly the proper normal slice integral $Q_{S}$ and the boundary term is the net outward entropy flux $Φ_{S} (B)$ . In the length-coordinate convention of [F1], these charges and the integrated production have units $J K^{- 1}$ . Hence the stated finite balance holds, conditional on the current draft theorem's exact interface.

3.1A1step 1.1step 2.1algebra∎

If $Φ_{S} (B) = 0$ , step 2.1 and step 1.1 give $Q_{S} (x_{+}^{0}) - Q_{S} (x_{-}^{0}) \geq 0$ . If also $Π$ vanishes throughout $K$ , the integral in step 2.1 is zero and the endpoint charges are equal. No monotonicity of the charge inside an open region is claimed when the net outward flux is nonzero.

Conditional finite entropy balance with explicit boundary flux

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources